Morse flow categories as exit path categories
Abstract
We prove that the topological flow category ℳ arising from a Morse-Smale pair (f,ξ) on a smooth closed manifold X is equivalent, as an ∞-category, to Lurie's ∞-category SingA(X) of exit paths in X with respect to the stratification by the stable manifolds of ξ. The objects of ℳ are the critical points of f, and for every pair of critical points, the space of morphisms of ℳ between these is the space of possibly broken trajectories of ξ connecting them; it can be identified up to homotopy with the space of unbroken ones. The latter maps naturally to the space of exit paths connecting these critical points; we prove this map to be a weak homotopy equivalence. Then, we combine these ingredients with several others to construct a zigzag of equivalences between the homotopy coherent nerve of ℳ, denoted 𝒩(ℳ), and SingA(X). The n-simplices of 𝒩(ℳ) are homotopy coherent diagrams of n composable morphisms of ℳ; we introduce the notion of unbroken diagram, yielding an ∞-subcategory of 𝒩(ℳ), which we refer to as the flow coherent nerve of ℳ. The simplices of the latter give rise to stratified maps out of a family of stratified cubes, into X. We organize this family into a functor from the category of finite ordered sequences of critical points, to the category of A-stratified topological spaces, and we prove a comparison result with the usual stratified geometric realization functor. We finally use a theorem of Tanaka that associates a functor to a semi-simplicial map between ∞-categories that satisfies certain conditions. Our theorem has implications regarding constructible sheaves and the description of homotopy types in terms of flow categories.
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BibTeX
@misc{fourel2026morse,
title = {{Morse flow categories as exit path categories}},
author = {Colin Fourel},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.27112},
eprint = {2605.27112},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.27112},
}
RIS
TY - UNPB
TI - Morse flow categories as exit path categories
AU - Colin Fourel
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2605.27112
UR - https://arxiv.org/abs/2605.27112
AB - We prove that the topological flow category ℳ arising from a Morse-Smale pair (f,ξ) on a smooth closed manifold X is equivalent, as an ∞-category, to Lurie's ∞-category Sing_A(X) of exit paths in X with respect to the stratification by the stable manifolds of ξ. The objects of ℳ are the critical points of f, and for every pair of critical points, the space of morphisms of ℳ between these is the space of possibly broken trajectories of ξ connecting them; it can be identified up to homotopy with the space of unbroken ones. The latter maps naturally to the space of exit paths connecting these critical points; we prove this map to be a weak homotopy equivalence. Then, we combine these ingredients with several others to construct a zigzag of equivalences between the homotopy coherent nerve of ℳ, denoted 𝒩(ℳ), and Sing_A(X). The n-simplices of 𝒩(ℳ) are homotopy coherent diagrams of n composable morphisms of ℳ; we introduce the notion of unbroken diagram, yielding an ∞-subcategory of 𝒩(ℳ), which we refer to as the flow coherent nerve of ℳ. The simplices of the latter give rise to stratified maps out of a family of stratified cubes, into X. We organize this family into a functor from the category of finite ordered sequences of critical points, to the category of A-stratified topological spaces, and we prove a comparison result with the usual stratified geometric realization functor. We finally use a theorem of Tanaka that associates a functor to a semi-simplicial map between ∞-categories that satisfies certain conditions. Our theorem has implications regarding constructible sheaves and the description of homotopy types in terms of flow categories.
ER -